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By employing the $1/N$ expansion, we compute the vacuumenergy~$E(\delta\epsilon)$ of the two-dimensional supersymmetric (SUSY)$\mathbb{C}P^{N-1}$ model on~$\mathbb{R}\times S^1$ with $\mathbb{Z}_N$ twistedboundary conditions to the second order in a SUSY-breakingparameter~$\delta\epsilon$. This quantity was vigorously studied recently byFujimori et\ al.\ using a semi-classical approximation based on the bion,motivated by a possible semi-classical picture on the infrared renormalon. Inour calculation, we find that the parameter~$\delta\epsilon$ receivesrenormalization and, after this renormalization, the vacuum energy becomesultraviolet finite. To the next-to-leading order of the $1/N$ expansion, wefind that the vacuum energy normalized by the radius of the~$S^1$, $R$,$RE(\delta\epsilon)$ behaves as inverse powers of~$\Lambda R$ for~$\Lambda R$small, where $\Lambda$ is the dynamical scale. Since $\Lambda$ is related tothe renormalized 't~Hooft coupling~$\lambda_R$ as~$\Lambda\sime^{-2\pi/\lambda_R}$, to the order of the $1/N$ expansion we work out, thevacuum energy is a purely non-perturbative quantity and has no well-definedweak coupling expansion in~$\lambda_R$.

Vacuum energy of the supersymmetric $\mathbb{C}P^{N-1}$ model on $\mathbb{R}\times S^1$ in the $1/N$ expansion